Posts Tagged ‘chemical potential’

I’d intended to rework the exam problems over the summer and make that the last update to my stat mech notes. However, I ended up studying world events and some other non-mainstream ideas intensively over the summer, and never got around to that final update.

Since I’m starting a new course (condensed matter) soon, I’ll end up having to focus on that, and have now posted a final version of my notes as is.

Here’s part of a problem from our final exam. I’d intended to redo the whole exam over the summer, but focused my summer study on world events instead. Perhaps I’ll end up eventually doing this, but for now I’ll just post this first part.

Question: Large volume Fermi gas density (2013 final exam pr 1)

Write down the expression for the grand canonical partition function of an ideal three-dimensional Fermi gas with atoms having mass at a temperature and a chemical potential (or equivalently a fugacity ). Consider the high temperature “classical limit” of this ideal gas, where and one gets an effective Boltzmann distribution, and obtain the equation for the density of the particles

by converting momentum sums into integrals. Invert this relationship to find the chemical potential as a function of the density .

Hint: In the limit of a large volume :

Answer

Since it was specified incorrectly in the original problem, let’s start off by verifing the expression for the number of particles (and hence the number density)

Moving on to the problem, we’ve seen that the Fermion grand canonical partition function can be written

Question: Low temperature Fermi gas chemical potential

In class, we assumed that was quadratic in as a mechanism to invert this non-linear equation. Without making this quadratic assumption find the lowest order, non-constant approximation for .

Answer

To determine an approximate inversion, let’s start by multiplying eq. 1.0.2 by to non-dimensionalize things

or

If we are looking for an approximation in the neighborhood of , then the LHS factor is approximately one, whereas the fractional difference term is large (with a corresponding requirement for to be small. We must then have

One can measure the specific heat in this Bose condensation phenomina for materials such as Helium-4 (spin 0). However, it turns out that Helium-4 is actually quite far from an ideal Bose gas.

Photon gas

A system that is much closer to an ideal Bose gas is that of a gas of photons. To a large extent, photons do not interact with each other. This allows us to calculate black body phenomina and the low temperature (cosmic) background radiation in the universe.

An important distinction between a photon sea and some of these other systems is that the photon number is actually not fixed.

Photon numbers are not “conserved”.

If a photon interacts with an atom, it can impart energy and disappear. An excited atom can emit a photon and change its energy level. In a thermodynamic system we can generally expect that introducing heat will generate more photons, whereas a cold sink will tend to generate fewer photons.

We have a few special details that distinguish photons that we’ll have to consider.

spin 1.

massless, moving at the speed of light.

have two polarization states.

Because we do not have a constraint on the number of particles, we essentially have no chemical potential, even in the grand canonical scheme.

Writing

Our number density, since we have no chemical potential, is of the form

Observe that the average number of photons in this system is temperature dependent. Because this chemical potential is not there, it can be quite easy to work out a number of the thermodynamic results.

Photon average energy density

We’ll now calculate the average energy density of the photons. The energy of a single photon is

so that the average energy density is

Mathematica tells us that this integral is

for an end result of

Phonons and other systems

There is a very similar phenomina in matter. We can discuss lattice vibrations in a solid. These are called phonon modes, and will have the same distribution function where the only difference is that the speed of light is replaced by the speed of the sound wave in the solid. Once we understand the photon system, we are able to look at other Bose distributions such as these phonon systems. We’ll touch on this very briefly next time.

Disclaimer

Fermi gas

Review

Continuing a discussion of [1] section 8.1 content.

We found

With no spin

Fig 1.1: Occupancy at low temperature limit

Fig 1.2: Volume integral over momentum up to Fermi energy limit

gives

This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intended. We see that both achieve the same result}, where

Moving on

with

this gives

Over all dimensions

so that

Again

Example: Spin considerations

{example:basicStatMechLecture16:1}{

This gives us

and again

}

High Temperatures

Now we want to look at the at higher temperature range, where the occupancy may look like fig. 1.3

Fig 1.3: Occupancy at higher temperatures

so that for large we have

Mathematica (or integration by parts) tells us that

so we have

Introducing for the thermal de Broglie wavelength,

we have

Does it make any sense to have density as a function of temperature? An inappropriately extended to low temperatures plot of the density is found in fig. 1.4 for a few arbitrarily chosen numerical values of the chemical potential , where we see that it drops to zero with temperature. I suppose that makes sense if we are not holding volume constant.

Fig 1.4: Density as a function of temperature

We can write

or (taking (and/or volume?) as a constant) we have for large temperatures

The chemical potential is plotted in fig. 1.5, whereas this function is plotted in fig. 1.6. The contributions to from the term are dropped for the high temperature approximation.

Disclaimer

Last time we found that the low temperature behaviour or the chemical potential was quadratic as in fig. 1.1.

Fig 1.1: Fermi gas chemical potential

Specific heat

where

Low temperature

The only change in the distribution fig. 1.2, that is of interest is over the step portion of the distribution, and over this range of interest is approximately constant as in fig. 1.3.

Fig 1.2: Fermi distribution

Fig 1.3: Fermi gas density of states

so that

Here we’ve made a change of variables , so that we have near cancelation of the factor

Here we’ve extended the integration range to since this doesn’t change much. FIXME: justify this to myself? Taking derivatives with respect to temperature we have

With , we have for

Using eq. 1.1.4 at the Fermi energy and

we have

Giving

or

This is illustrated in fig. 1.4.

Fig 1.4: Specific heat per Fermion

Relativisitic gas

Relativisitic gas

graphene

massless Dirac Fermion

Fig 1.5: Relativisitic gas energy distribution

We can think of this state distribution in a condensed matter view, where we can have a hole to electron state transition by supplying energy to the system (i.e. shining light on the substrate). This can also be thought of in a relativisitic particle view where the same state transition can be thought of as a positron electron pair transition. A round trip transition will have to supply energy like as illustrated in fig. 1.6.

Fig 1.6: Hole to electron round trip transition energy requirement

Graphene

Consider graphene, a 2D system. We want to determine the density of states ,